The highest barrier that a projectile can clear is 15.8 m, when the projectile is launched at an angle of 13.4° above the horizontal. What is the projectile's launch speed?

"vertical" kinetic energy = max potential energy change.

(Horizontal kinetic energy does not change)

(Vo sin13.4)^2/2 = 2g H

Solve for Vo.

To find the projectile's launch speed, we can use the principles of projectile motion and break it down into its horizontal and vertical components.

Let's break down the given information:
- The highest barrier that the projectile can clear is 15.8 m.
- The projectile is launched at an angle of 13.4° above the horizontal.

First, we can find the time taken by the projectile to reach its maximum height. At the highest point of its trajectory, the vertical component of velocity becomes zero. We can use the following formula to calculate the time:

Vf = Vi + a * t

Here, Vf is the final vertical velocity, Vi is the initial vertical velocity, a is the vertical acceleration (which is equal to -9.8 m/s^2 due to gravity), and t is the time taken.

At the highest point, the projectile's final vertical velocity is 0 m/s, and the initial vertical velocity can be found using the launch speed and the angle:

Vi = V * sinθ

where V is the launch speed and θ is the launch angle.

Using the formula, we can solve for t:

0 = Vi - 9.8 * t

t = Vi / 9.8

Next, we can find the time of flight, which is the total time the projectile remains in the air. Since the projectile goes up and then comes back down, the total time is twice the time taken to reach the maximum height:

Time of flight = 2 * t

Now, we can find the horizontal distance traveled by the projectile. The horizontal velocity remains constant throughout the motion:

Vh = V * cosθ

where Vh is the horizontal velocity.

Using the formula, we can calculate the horizontal distance:

Horizontal distance = Vh * Time of flight

Finally, we can use the given information to find the launch speed:

15.8 m = Horizontal distance
= V * cosθ * 2 * t

Substituting t = Vi / 9.8:

15.8 m = V * cosθ * 2 * (Vi / 9.8)

Now, we can plug in the values we know:
- θ = 13.4° (angle above the horizontal)
- Vi = V * sinθ (vertical component of the velocity)

We have two equations with two unknowns (V and Vi). By solving these equations simultaneously, we can find the launch speed.